VARIABLE STARS 
195 
/ 
Now 
wtR _ n 2 R 3 _ 4tt 2 3 3t t p c 
~9o ~ QM ~ IP = IPGft pf, 
- 12-2ya 
(131-3), 
by (130-3) introducing the value Pe / Pm = 54-36. 
h or the two groups of Cepheids Nos. 5—9 and 10—14 we found 
(ya)* = -342, -302. 
Substituting in (131-2), we see that at the boundary is between — AP X 
and — £P x . Since the pressure cannot become negative P x must not 
exceed unity. Hence there is an upper limit to £ x or SP/P between i and £. 
Although the observed values of 8P/P in Table 25 do not reach so high 
a limit, it seems possible that the vanishing of the pressure is effective 
in setting the limit to the amplitude attained in Cepheid pulsation and 
that the more typical Cepheids reach this limit. It has been explained 
that the values of 8P/P are probably systematically too low since the 
spectroscopic determination of SP refers to the integrated light of a hemi 
sphere. Allowing for this the amplitude generally attained seems to be 
about half the theoretical limit. The discrepancy may well be due to certain 
imperfections in the theory. When P x = 1 it is clearly illegitimate to 
neglect the squares of the amplitudes as we have done; a correction on 
account of this is necessary, but I do not think this is the whole cause of 
the difference. It is not sufficient that the total pressure P should remain 
positive; both p Q and p R must be positive. In the outermost part of the 
star p R and p G become out of phase with one another. This phase-difference 
is not shown in our theoretical equations because it is a result of the 
failure of the adiabatic approximation in this region; but we supply the 
gap by our observational knowledge. The critical time is when the star 
is at its greatest expansion. At that time we know from observation that 
the star is near its mean luminosity. Since the outflowing stream of 
radiation has its mean intensity it seems permissible to assume that p R 
has its mean value which for an average Cepheid is about |P 0 . Then since 
p G t is not negative we must have P 0 + SP > |P 0 which requires that 
the amplitude of P x shall not exceed By (131-2) the corresponding 
limit of is to in good agreement with observation. This argument 
depends on a patched-up treatment of the non-adiabatic region which 
may be fallacious, and it is put forward only as a suggestion*. 
* I believe that some years ago, when closely engaged with this branch of the 
subject, I came to the conclusion that the whole argument given in this section 
was fallacious; but I cannot remember the reasons, and do not now see the flaw, if 
any. In Monthly Notices, 79, p. 22 (1918) I stated that the discussion would appear 
in Part II of the paper, but for reasons now forgotten withheld it when Part II was 
published. 
Hence